We now study the Decomposition Theorem for \(h \colon Z \rightarrow N_{pq}\) . Proper base change and the fact that \(q\)  is a finite map imply that \(Rh_*\mathbb {Q}_Z = Re_*\mathbb {Q}_{\widetilde{K}} \oplus Re_*(j_{W*} {L}_W)\) . Lemma REF  implies that there is an isomorphism \(Re_*\mathbb {Q}_{\widetilde{K}} \simeq {I}_{N_{pq}}^+\) . To finish the proof, note that proper base change for \(e\)  implies there is an isomorphism \(j_{pq}^*Re_*(j_{W*} {L}_W) \simeq Re_*{L}_{\widetilde{K}_{U_{pq}}}\) . This isomorphism, coupled with the projection formula and Deligne's Theorem (see Theorem 1.5.3 in the survey [1]}), imply that the strings \({I}_{N_{pq}}^+\)  and \({I}_{N_{pq}}^-\)  are direct summands appearing in the Decomposition Theorem for \(Rh_*\mathbb {Q}\) . Considering the description of fibers of \(h\)  over \(N(3)\)  and \(NR\)  points above gives the skyscraper contributions.

